Massachusetts Institute of Technology Department of Physics 8.01T Fall 2004 Study Guide Final Exam The final exam will consist of a two sections. Section 1: multiple choice concept questions. There may be a few concept questions on time and special relativity but no analytic questions. Section 2: analytic problems with some concept questions requiring written responses. The analytic questions will be divided into part A covering the material since the third test. Part B will cover problems from the entire year. Examples of section 2 questions are given below. Test questions will not use numbers so you do not need a calculator. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines Problem 1 Energy Transformation, Specific Heat and Temperature Suppose a person of mass m = 6.5×101 kg is running at a speed v = 3.8 m s and is expending 9.45×102 W of power during a 1.0 101 km workout. Suppose the runner×converts 20% of the internal energy change into mechanical work. The rest of the energy goes into heat. If the specific heat of the runner is c = 4.19 ×103 J kg − K , how much would the body temperature rise after running the10 km ? Problem 2 Kinetic Theory An ideal gas has a density of 1.78 kg/m3 is contained in a volume of 44.8 x 10-3 m3 . The temperature of the gas is 273 K. The pressure of the gas is 1.01 x 105 Pa. The gas constant -1R = 8.31 J K-1 ⋅ mole .⋅ a) What is the root mean square velocity of the air molecules? b) How many moles of gas are present? c) What is the gas? d) What is the internal energy of the gas?Problem 3: Carnot Cycle of an Ideal Gas In this problem, the starting pressure P and volume V of an ideal gas in state a, area agiven. The ratio RV = V /V > 1 of the volumes of the states c and a is given. Finally a c a constant γ= 5/3 is given. You do not know how many moles of the gas are present. a) Read over steps (1)- (4) below and sketch the path of the cycle on a PV plot on the−graph below. Label all appropriate points. (1) In the first of four steps, a to b , an ideal gas is compressed from V to Vb while noaheat is allowed to flow into or out of the system. The compression of the gas raises the temperature from an initial temperature T and to a final temperature T . During this1 2process the quantity PV γ= constant , where γ= 5/3. a) What is the pressure P and volume Vb of the state b of the gas after the bcompression is finished? b) What is the change in internal energy of the gas during this change of state? c) What is the work done by the gas during this compression? (2) The gas is now allowed to expand isothermally from b to c , from volume V tobvolume V .cd) Express the work done by the gas in this process Wcb and the amount of heat Qcbthat must be added from the heat source at T2 in terms of P , V , T , T , and V .a a 2 1 cIs this heat positive or negative? Explain whether it is added to the system or removed. e) What is the pressure P of the gas after the expansion is finished? c(3) When the gas has reached point c is expands from V to Vd while no heat is allowedcto flow into or out of the system. The expansion of the gas lowers the temperature and pressure from an initial temperature T2 to a final temperature T . During this process the1quantity PV γ=constant . f) What is the pressure P and volume Vd of the state d of the gas after the dexpansion is finished? g) What is the change in internal energy of the gas during this change of state? h) What is the work done by the gas during this expansion? (4) The gas is now compressed isothermally from d to a at constant T from volume Vd1back to V .aQi) Find the work done by the system on the surroundings Wad and the amount of heat ad that flows between the system and the surroundings. Are these quantities positive or negative? Explain whether heat is added to the system or removed from the heat source at T1. Total Cycle: j) What is the total work Wcycle done by the gas during this cycle? k) What is the total heat Qcycle ( from T) drawn from the higher temperature heat 2 source during this cycle? l) What is the efficiency of this cycle εmax =Wcycle / Qcycle ( from T ) ? 2 Problem 4 Heat pump A reversible heat engine can be run in the other direction, in which case it does negative work Wcycle on the world while “pumping” heat Qcycle (into T) into a reservoir at an upper2 temperature, T , from a lower temperature, T . The heat gain of this cycle, defined to be2 1gQcycle (into T )/Wcycle = (1/ ε )≡ 2 max where ε= (TT )/ T is the maximum thermodynamic efficiency of a heat engine. max 2 −1 2 The refrigerator performance is defined to be KQcycle ( from TWcycle = T /(T − T )≡ 1)/ 1 2 1 Consider that you have a large swimming pool and plan to heat your house with a heat pump that pumps heat from the pool into your house. A large plate in the water will remain at 0 oC due to the formation of ice. You pick T2 to be 50 oC , which will be the temperature of the (large) radiators used to heat your house. Assume that your heat pump has the maximum efficiency allowed by thermodynamics. a) What is the heat gain and the refrigerator performance for this cycle? Be careful to use units of Kelvin for temperature. b) If your house formerly burned 1200 gallons of oil in a winter (at $2.00/gallon), how much will the electricity cost (at $0.10 per kilowatt-hour) to replace this heat using 8 ⋅the heat pump? A gallon of oil has mass 3.4 kg and contains 1.4 ×10 J gal-1. c) The ice cube that appears in your pool over the winter will be how many meters on 6each side? (It takes 3.35×10 J to melt one kg of ice; it takes up this much heat when 3 ⋅freezing.) The density of ice is 0.931×10 kg m-3 . This would be great for cooling your house in the summer – even if the pool warmed up enough to swim in it, you could still cool your house by running the heat pump in reverse as an air conditioner! More practically, you might be able to use ground water (and the dirt around it) as the heat sink.⋅ ⋅Part Two: Earlier Material Problem 1: (Momentum and Impulse) A superball of m1 = 0.08kg , starting at rest, is dropped from a height falls h0 = 3.0m above the …
View Full Document
Unlocking...